Structural damage detection based on improved frequency change ratio

Frequency Change Ratio (FCR) based damage detection methodology for structural health monitoring (SHM) is analyzed in detail. The effectiveness of damage localization using FCR for some slight damage cases and worse ones are studied on an asymmetric planar truss numerically. Disadvantages of damage detection using FCR in practical application are found and the reasons for the cases are discussed. To conquer the disadvantages of FCR, an Improved Frequency Change Ratio (IFCR) based damage detection method which takes the changes of mode shapes into account is proposed. Verification is done in some damage cases and the results reveal that IFCR can identify the damage more efficiently. Noisy cases are considered to assess the robustness of IFCR and results indicate that the proposed method can work well when the noise is not severe.


Introduction
Structures may deteriorate and suffer from accumulated damage under the external loads during their service life. When damages occur, the performance of the structure will degrade. It is of great significance to identify the damaged parts of the structure in a timely manner so that we can diminish the probability of accident, enhance the stability and prolong the service life.
Vibration based structural damage detection become an area of intense investigation and gain a lot of research improvement around these decades [1]. Structural damage will induces some changes in the modal parameters of the structure [2], such as the natural frequencies, mode shapes, damping ratios etc. Salawu [3] reviewed the methodologies using natural frequencies as diagnostic parameters for structural damage detection.
In 1968, Fox and Kapoor [4] derived the rates of change of eigenvalues and eigenvectors which traced back to the investigations of Lord Rayleigh. The derivation indicates that the ratio of frequency changes of two different modes is merely a function of damage location, which could be used for damage detection [5].
In this paper, the Frequency Change Ratio (FCR) based damage detection method and its application on an asymmetric planar truss is described in detail and the defects of FCR and the possible reasons are discussed. And based on this, some improvements are proposed to form an Improved Frequency Change Ratio(IFCR) damage detection method. The planar truss is applied to investigate the effects of the proposed IFCR method numerically.
i ij As shown in Eq. (3), when there is a single damage occurs on the structure, ij FCR is independent of the damage severity and only related to the damage position. Hence, damage location can be realized through identifying the ratio of natural frequency changes.
Damages may change the natural frequencies of the structure. A judgment index Er is established for the judgement of damage location, which is expressed as A two-dimensional planar truss as shown in Fig.1 is used to investigate the capability of the FCR damage detection method. The Young's modulus E of the truss elements is 2.0×10 8 Pa and the density  is 7850kg/m 3 . All bars of the truss have uniform cross sections A =7.854×10 -5 m 2 . The total length of the truss is 6m and the length of each horizontal bar is 2m while the length of each inclined ones is 2 m.       From Fig.3 we can see that the first member is the damaged bar visibly after filtering process. Therefore, the filtering procedures are adopted in the following papers for damage identification.
For the supposed damage case, five natural frequencies before and after damage are listed in Tab.1.  Comparing Fig.3 with Fig.4, the damage recognition capability of worse damage cases is not as good as the slight damage case. From Fig.4(b) we can see that the Er value of the 2 nd member (which is 7.26) is close to the value of 1 st member (which is 2.04), and this may decrease the separability of the judgments. For the case shown in Fig.4(c), the Er value of the 1 st member, 2 nd member and 10 th member are close to each other, then it will produce more difficulties for the damage identification. In Fig.4(d) the smallest Er value is located on the 2 nd member(which is 6.11) and it would be misjudged as the damaged one. Same mistakes would be done when the 8 th member suffers from severe damage as shown in Fig.4(g) and Fig.4(h). Tab.3 shows the concrete relationship between damage severity and MSSR is complex, and generally, there is an approximate trend that the mode shapes will change more as the damage severity increases. Additionally, some mode shape shift ratios are huge under certain damage cases.         The more the damage severity increases, the more the frequencies change. This phenomenon means that more serious damage is beneficial for FCR damage detection. But the mode shapes will shift more at the same time and it will inhibit the validity of the damage detection method.

Improvement method for FCR
In the light of analysis in section 2.2.2, the mode shape shift could not be neglected. We can get considering Eq. (1) and noting that . Neglect the change of mode shape i   , then Eq. (6) will become the Eq. (2). Different from the FCR method, the high order terms ( are not neglected as usual, and actually, Eq. (6) is exactly the equation of the parameters without any errors. Conveniently, we call this method the Improved Frequency Change Ratio (IFCR) method. Utilizing the IFCR, damage detection of the 1 st member 5% damage is shown as shown in Fig.5.   When the damage is slight, the influence of mode shape caused by damage will be small, the IFCR can also identify the damage location successfully as shown in Fig.5.

Damage localization using IFCR
For some worse damage cases, three damage cases are chosen to verify the localization ability of the proposed IFCR damage detection method. Damage case I represents the 1 st member occurs 30% damage severity, damage case II represents the 1 st member suffers from 70% damage severity, and the damage case III means the 8 th member has 70% damage severity. Using the proposed IFCR method, and applying the filtering process, Er distributions for damage localization are shown in Fig.6. The damaged members can be distinguished apparently in Fig.6 and hence IFCR is a better damage detection method.

Noise affections
As we all known, frequencies can be measured with a high precision while the mode shapes may be influenced more severely under the present experimental techniques. Some random noises are added to the calculated frequencies and mode shapes. The contaminated frequency with noise can be gained using the following equation: f'=f(1+×rand[-1,1]) (9) where f is the calculated frequency, f' is the contaminated frequency with noise,  denotes the noise level(e.g. 0.05 relates to a 5% noise level) and rand is a random number in the range [-1,1]. Like the frequency, mode shape with noise can also be expressed like Eq. (9).    According to this, a serious of noises listed in Tab.5 is added in the natural frequencies and mode shapes for damage identification.  In order to reveal the typical influence of the noises, damage case I is selected to be mixed with the noises. The IFCR method is adopted to identify the damaged location and the results are shown in Fig.7.    As shown in Fig.7, the noises may do harmful impact to the damage identification to some severity. IFCR method can identify the damage location clearly even though the frequencies and mode shapes are suffered from 5% and 10% noises respectively. The recognition effect of damage case I under noise case I reflects that IFCR has a strong antinoisy ability. When the noises increase, the identification ability will decline such as the noise case III, misjudgment may be done in this situation.

Conclusions
Frequency change ratio based damage detection method is discussed in detail in this paper. The identification for a planar truss shows that the FCR method may have some disadvantages for recognizing some slight damage cases and the worse damage cases in practical situations. Accordingly, an improved frequency change ratio based damage detection method IFCR is proposed. Damage localization effects in noise-free and noise cases are studied and the results show that the proposed IFCR method can identify the damaged member accurately when the noise is not very severe.